The effect to Special Summon cannot be applied during the Damage Step.
Previously Official Rulings
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
The victory check for "Last Turn" is an effect that must be resolved by the player who activated it. The turn player still has priority to activate and resolve any other effects first. You cannot chain to the victory check of "Last Turn".
"Waboku" can be activated before "Last Turn", or be chained to the activation of "Last Turn", and in either case the effect of "Waboku" will prevent the activator’s monster from being destroyed. This can lead to a DRAW.
Fear from the Dark: If "Fear from the Dark" is sent to the Graveyard by "Last Turn"'s effect, "Fear from the Dark" is Special Summoned to the field before the attack of "Last Turn" happens.
Fiend Skull Dragon: Cards such as..."Last Turn" will all apply their effects to "Fiend Skull Dragon" because these cards do not target.
Frozen Soul: While "Frozen Soul" is in effect, you cannot activate "Last Turn" in the same turn because "Last Turn" includes conducting a Battle Phase.
Gyroid: If "Gyroid" is attacked during a Special Battle Phase created by the effect of "Last Turn", and is then attacked again during the regular Battle Phase, it will be destroyed because its effect was already applied once that turn (during the Special Battle Phase).
Magical Scientist: If "Jowgen the Spiritualist" is on the field, you cannot activate the effect of "Magical Scientist" (so you cannot even pay the cost to try and activate "Last Turn", for example).
Night Assailant: When "Night Assailant" is sent to the Graveyard by the effect of "Last Turn", FIRST resolve "Last Turn"'s effect of sending all cards on the field and in the hands to the Graveyard, SECOND activate and resolve the effect of "Night Assailant", and THIRD Special Summon a monster by the effect of "Last Turn".
Obnoxious Celtic Guard: "Obnoxious Celtic Guard"'s effect works for a battle caused by "Last Turn", so if it battles a monster with ATK 1900 or higher, neither is destroyed and the Duel is a draw.
Tyrant Dragon: You may activate "Last Turn" and select your "Tyrant Dragon" as the monster to keep, because "Last Turn" does not target.
Wall of Revealing Light: You can chain "Last Turn" to your activation of "Wall of Revealing Light" as long as "Last Turn"'s conditions are then correct (1000 or less Life Points).
↑ 1.01.1The latest print of "Dark Magician of Chaos" uses Problem-Solving Card Text, which makes it clear that its effect activates during the End Phase of the turn it was Normal or Special Summoned, not when it is Summoned. This difference is due to a major change in the functionality of "Dark Magician of Chaos".